The idea that teachers can use the principle of Gillick competence to allow children to socially transition in schools is clearly misguided. Gillick is an assessment undertaken on a case by case basis to children under the age of 16. Looking at the NSPCC information (quoted below) linked in the OP it indicates the process that one would need to go through for each child to assess their competency. You cannot make a blanket decision that it is OK for children of a particular age to socially transition (this was confirmed in the Bell case). The idea that teachers have the training, knowledge, expertise and time to undertake such an assessment is totally deluded.
Assessing Gillick competence
There is no set of defined questions to assess Gillick competency. Professionals need to consider several things when assessing a child's capacity to consent, including:
The child's age, maturity and mental capacity
Their understanding of the issue and what it involves - including advantages, disadvantages and potential long-term impact.
Their understanding of the risks, implications and consequences that may arise from their decision.
How well they understand any advice or information they have been given
Their understanding of any alternative options, if available.
Their ability to explain a rationale around their reasoning and decision making.
Remember that consent is not valid if a young person is being pressured or influenced by someone else.
[This is a massive risk in schools]
Children's capacity to consent may be affected by different factors, for example stress, mental health conditions and the complexities of the decision they are making. The same child may be considered Gillick competent to make one decision but not competent to make a different decision.
If you don't think a child is Gillick competent or there are inconsistencies in their understanding, you should seek consent from their parents or carers before proceeding.